Information Structure and Comparative Statics in Simple
Global Games
Michal Szkup
University of British Columbia
September 20, 2014
Very Preliminary and Incomplete
Abstract
This paper analyzes the e¤ect of changing information structure in simple global games.
I provide a complete characterization of the e¤ect of private and public precision on the
fundamental threshold. Interestingly, I …nd that away from the limiting case of in…nitely
precise signals, the threshold can be non-monotonic in both precision of public and private
information. I provide intuition behind this result and argue that this has important policy
consequences.
1
Introduction
Global games are a popular way of modeling coordination problems such as bank runs, currency crisis or sovereign debt crises. This popularity stems from the fact that while underlying
coordination game features multiple equilibria, the global games, by distorting information
structure, restores uniqueness of equilibrium. The key idea of this approach is that incomplete
information impedes coordination since in this case agents have to rely on their private information when choosing their actions. As shown …rst by Carlson and Damme (1993), in the
limit, as private information become in…nitely precise, the set of equilibria becomes a singleton.
Moreover, under mild conditions, this equilibrium is independent of the initial perturbation
applied to the underlying complete information game (Frankel, Morris, and Pauzner (2003)).
While the limiting result is important, in reality individuals rarely have an access to extremely precise information. Thus, from an applied perspective is important to understand
how the unique global game equilibrium is a¤ected by a change in the precision of private
and public information available to agents. Indeed, there is now a large literature that studies
how changes in information structure in di¤erent types of coordination games with incomplete
information a¤ects equilibrium play (see for example Angeletos and Pavan (2007), Colombo,
Femminis, and Pavan (2012) or Iachan and Nenov (2014)).1 In this paper, I contribute to
1
See for example Rochet and Vives (2004), Morris and Shin (2004) or Iachan and Nenov (2014) for global
games; Angeletos and Pavan (2007) or Colombo, Femminis, and Pavan (2012) for the analysis in the quadraticGaussian setups (often referred to as beauty contest models).
1
this literature by providing a full characterizing the comparative statics result with respect to
information parameters in global games away from the limit of perfectly informative signal.
I study what I call “simple global games”, i.e., binary global games where the payo¤
di¤erence between actions is a step function, agents hold a proper common prior, and the
information structure is either “Gaussian” or “uniform”. Such speci…cations of global games
has been widely used in an applied global games literature.2 I show that the comparative
statics with respect to precision of information, away from the limit, depend on three e¤ects
which I call a “mean e¤ect”, a “dispersion e¤ect” and an “aggregate e¤ect”. The “mean
e¤ect”captires the e¤ect that a change in the precision of information has on the mean of the
marginal agent’s posterior belief, i.e. of the agent who is indi¤erent between taking each of the
two actions. The “dispersion e¤ect”is in turn the e¤ect that a change in precision of public or
private information has on the posterior belief’s variance of the marginal agent. Finally, the
“aggregate e¤ect”is the change in proportion of agent receiving a signal less than the threshold
signal.
The above three e¤ects determine the behavior of the regime change threshold at a given
precision of private and public information. Using the above decomposition I fully characterize
how an incremental increase in the precision of private or public information a¤ects the unique
equilibrium away from the limit of perfectly informative signals. In particular, I provide
exhaustive conditions, that depend only on the exogenous parameters of the model, under
which a small increase in the precision of private or public information leads to an increase or
decrease in the probability of agents coordinating on the risky action. While it is important
to understand the local comparative statics with respect to parameters governing information
structure, I also analyze the global behavior of the regime change threshold, i.e. the evolution
of equilibrium as we continuously vary precision of information. I provide a full description
of the global behavior of equilibrium and, in particular, I characterize conditions under which
equilibrium threshold is a non-monotone function of information parameters. I …nd that when
the prior belief is low it is possible that the equilibrium threshold is …rst increasing and then
decreasing. Similarly, if the prior belief is high, it is possible that the equilibrium threshold is
…rst decreasing and then increasing. I show that these are the only types of non-monotonicity
that can arise in simple global games.
The above observations have potentially important implications for policy makers. In
particular, in the case where reducing the initial uncertainty (i.e., increasing precision of the
prior) is costly to policy makers, it might be optimal for the policy maker not to increase the
precision of information at all since a small increase can actually have detrimental e¤ect on
the economy. Similarly, the above conditions can be used to determine when the policy maker
should try to decrease initial uncertainty following an increase in the amount of information
available to agents and when to increase it.
In the …nal part of the paper I introduce public information focusing on the setup with
“Gaussian” noise. The results mentioned above are necessary building steps for analysis of
the e¤ect of the precision of an explicit public signal on equilibrium. In turns out that the
presence of public information substantially complicates the analysis and renders the approach
2
See for example Eisenbach (2013) or Rochet and Vives (2004) for applications to banking crises, Corsetti,
Guimaraes, and Roubini (2006) and Morris and Shin (2006) for soveriegn debt crisis, Morris and Shin (2004)
and Szkup (2013) for corporate debt crises, or Corsetti, Dasgupta, Morris, and Shin (2004) and Dasgupta (2007)
for currency crises just to mentions a few.
2
used to characterize comparative statics in the absence of public signal inapplicable. Thus, in
this part of paper I utilize a di¤erent approach. This approach is based on the observation
that the equilibrium regime threshold, as a function of public signal realization, exhibits a
special form of an asymmetry Using this observation I provide a complete characterization
of an incremental increase in the precision of private and public information in this extended
setup. Finally, I show numerically, that the non-monotonicity of the equilibrium regime change
threshold carries over to the case with public information.
This paper contributes to a large and growing literature on the theory of global games and,
more broadly, coordination games with incomplete information. The global games have been
initially developed by Carlson and Damme (1993) and then extended by Frankel, Morris, and
Pauzner (2003) to wider range of setups.3 Since then the global games have been extended
along many dimensions and the robustness of the uniqueness result have been analyzed extensively. I contribute to this literature by characterizing the comparative statics with respect to
information parameters in simple global games.
I am not the …rst person to consider the comparative statics with respect to the precision
in the simple global games. The early work on this issue includes Metz (2002) and Bannier and
Heinemann (2005). Bannier and Heinemann (2005) analyze the optimal level of public and
private precision in a simpli…ed version of currency attack model developed by Morris and Shin
(1998). They were the …rst one to report that the equilibrium threshold may not be monotone
with respect to the precision of public information. My paper extends their results in several
dimensions. First of all, I providing exhaustive conditions under which equilibrium regime
change threshold is non-monotone in the precision of both the prior belief as well as private
signal under no assumptions on parameters. Second, I provide a detailed intuition behind
the forces that lead to this monotonicity that applies beyond the Gaussian noise structure.
Finally, I also consider a framework with both private and public signals. More recently
Iachan and Nenov (2014) provide a general analysis of comparative statics with respect to
the informativeness of private signals in global games. In contrast to this paper the allow the
payo¤s to be functions of the underlying economic parameter. While the setup they analyze is
more general than the setup in this paper, the added complexity requires them to focus either
on the environment with improper prior or a limiting case where the precision of private signal
tends to in…nity.4
2
Framework
There is a continuum of agents indexed by i with i 2 [0; 1]. The economy is characterized by
a parameter , which represents the strength of the economy’s fundamentals, that is a higher
represent stronger fundamentals. I assume that R in distributed according to a distribution
G. Let ai be agent i’s action and denote by A = ai di the aggregate action in the economy.
The utility function of each agent is given by u (ai ; A; ) : Before deciding which action to take
each agent observes a private signal xi . Finally, assume that the economy can be in one of the
3
See also the recent paper by Oury (2014) for the extension to the multi-dimensional global games.
The paper takes the coordination stage as given and treats changes in private and public information as
comparative statics exercises. Szkup and Trevino (2009), building on the results in this paper, analyze an
environment where agents’can explictly choose the precision of private information choices.
4
3
two regimes, R 2 f0; 1g and initially is in regime 0. The regime changes if f ( ; A)
stays the same if f ( ; A) < 0.
2.1
0 and
Simple global game
I call a global game “simple” if the payo¤ and information structure satisfy the following
assumptions on payo¤ and information structure.
Assumption 1 (Payo¤ Structure) The payo¤ structure satis…es:
ai 2 fa1 ; a2 g
u (a2 ; A; )
H if f (A; ) 0
L if f (A; ) < 0
u (a1 ; A; ) =
f ( ; A) = p1 A + p2
1 with p1 ; p2 > 0
Assumption 2 (Information Structure) The information structure satis…es:
N
unif [
;
1
and xi = +
;1 +
x
1=2
"i , "i
N (0; 1), or
] and xi = + "i , "i
unif [
x; x]
Thus, I de…ne a simple global game as a game where the actions are binary, the di¤erence
between payo¤s from taking action 1 and action 2 is constant in each regime and the information
structure is either “Gaussian” or “uniform” A vast majority of global games fall under this
de…nition (e.g. see Morris and Shin (2003) or Veldkamp (2011)).
3
Equilibrium
It is well known that (under additional assumptions on the information structure) the model
has unique equilibrium (Morris and Shin (1998) and Morris and Shin (2004)). The next
proposition, stated without the proof, summarizes these conditions.
Proposition 1
1. In the case of Gaussian information structure, a simple game has unique
equilibrium as long as
1=2
x
>
p1
2
2. In the case of uniform information structure, a simple global game has unique equilibrium
as long as 2 x > .
In both cases the equilibrium is characterized by a pair of thresholds f ; x g such that the
regime changes i¤ >
and the agent chooses action 2 if he observes a signal xi > x .
The goal of the rest of this paper is to analyze how the equilibrium is a¤ected by the
particular features of the information structure such as the informativeness of the signals.
The equilibrium is determined by two equations. First, the payo¤ indi¤erence condition,
that says that if the regime changes if and only if
then an agent who received a critical
signal x is indi¤erent between acting, action a2 , and not acting, action a1 , i.e.
H Pr (
jx ;
x,
) = L Pr ( <
4
jx ;
x,
)
or, written more succinctly,
P ( ;x ;
x,
) = 0.
The second equilibrium condition is the critical mass condition which says that when fundamentals are equal to
then the mass of agents taking action a2 is exactly equal to the mass
of agents needed for the regime change:
p1 Pr (xi
x j ;
x,
)=1
M ( ;x ;
x,
)=0
p2
or, more succinctly as
In the remainder of the paper I investigate the properties of this equilibrium. In particular
I investigate how the threshold x and
changes with the precision of private and public
information captured by x and , respectively.
De…nition 1 Let
L
H+L .
Parameter 2 (0; 1), which can be interpreted as the relative attractiveness of acting, will
turn out to be the key parameter when determining the comparative statics results.
3.1
Preliminaries
Let be a parameter describing information structure. It could be the precision of private
or public information in the Gaussian information structure, or the parameter governing the
error term in the uniform information structure. However, for now we abstract from the exact
nature of this parameter and investigate how a change in any parameter of the information
structure a¤ects the equilibrium regime change threshold .
A change in a¤ect the equilibrium by changing the equilibrium conditions. To see that
recall that the equilibrium is determined by to equations:
M (x ;
; ) = 0
P (x ;
; ) = 0
Totally di¤erentiating these two equations and solving for d =d we obtain:
d
d
=
or
d
d
=
1
M P
Mx Px
Mx P
M Px
M
M
1
@
@h
@
+ @x
i
@
@x
,
@x
@
@x
M @
,
(1)
P
where the subscript on the square brackets in the numerator indicates the equation from which
the partial derivative was computed.
A change in information parameter potentially a¤ects the regime change threshold through
three channels: (1) the “mean e¤ect” - a change in the posterior mean of the marginal agent
(i.e., the agent who is indi¤erent between acting and not acting), (2) the “dispersion e¤ect”- a
5
change in the posterior variance of the marginal agent, and (3) the “aggregate e¤ect”- a change
in the mass of agents acting (or, in other word, a change in the probability that an agent receives
a signal higher than the threshold signal x for given ). The term [@ =@x ] [@x =@ ] captures
both the “mean e¤ect” and the “dispersion e¤ect”, while @ =@ captures the “aggregate
e¤ect”.5 Using this simple insight, I next determine the relative strength of these e¤ects when
varying x , the parameter that a¤ects the “precision”of private signals, and , the parameter
that a¤ects the “precision” of the prior belief.
4
4.1
A change in the precision of private information
Gaussian Information Structure: Local Behavior
I consider …rst the e¤ect of an increase in the precision of private signals for the case of Gaussian
Information Structure. In this case the payo¤ indi¤erence conditions is given by:
!
xx +
x+
1
(
x
+
)
=
1=2
Consider an increase in x . From the above equation it follows that the mean e¤ect, i.e. the
change in x implied by a change in the marginal agents’posterior mean keeping the posterior
variance ( x + ) 1=2 constant, is given by
@x
@ x
=
M ean
2
x
(
)
2
x
p
1
x+
1
( )
Thus we see that the mean e¤ect is negative when
is low and when is high. The intuition
behind that is simple. A change in private precision increases the weight of the threshold
signal in the marginal agents’posterior belief and decreases the weight of the prior. When
is low and is high the threshold signal is high x is high and thus an increase in x shifts
agent’s posterior belief up. Thus, agents who receive signals x becomes more con…dent that
the risky action will be successful and they switch to taking risky action. As a consequence
the threshold signal has to decrease.
In the similar fashion, we can …nd the “dispersion e¤ect”, i.e. the e¤ect that a change in
posterior variance, caused by varying x , has on x :
@x
@ x
=
Dispersion
1
1
p
2
x+
1
( )
We see that dispersion depends on and (precision levels x and ) but not on . To
understand this suppose that is high. Then agents are particularly worried of committing a
mistake and undertaking a risky action when the risky action will be unsuccessful since this will
be associated with a large negative payo¤. To avoid this, they sets x such that the threshold
is smaller than the mean of his posterior belief about . This implies that for signals that
agent does invest, the probability that investment is successful is high which compensates the
5
Under the conditions that ensure uniqueness the denominator is always positive when determining the e¤ect
of a change in the parameters of information structure it is enough to focus on the relative strength of the mean,
dispersion and aggregate e¤ect.
6
agent for the possibility of the costly bad outcome. But when x increases the tails of the
posterior distribution shrink. In particular, holding the posterior mean constant, when the
posterior mean is higher than , the probability mass below
decreases implying that the
possibility of making the costly error is smaller. This in turn encourages the agent to decrease
his threshold signal x . Analogous logic holds when is low.
The change in x a¤ects then
through the critical mass condition. The e¤ect of higher
x on
implied by the critical mass condition is captured by
1=2
x
@
=
@x
1+
x
1=2
x
1=2
x
x
1=2
x
implying that a higher x always leads to higher regime change threshold . Therefore, the
initial contribution of the mean and dispersion e¤ect to a change in
is given by
@ @x
=
@x @ x
1=2
x
"
x
1=2
x
1=2
x
1+
x
1=2
2
x
+ 21
)+ 2 p
x+
x
1
(
x
1
#
( )
x
Finally, I compute the aggregate e¤ect captured by @ =@
theorem in the critical mass condition we get
1=2
@
@
x
(x
x
1 x
=
2 1+
Using the implicit function
)
1=2
x
x
1=2
x
x.
x
1=2
x
The sign of the distance e¤ect depends on the relative position of x and . If x <
then
it follows that an increase in x increases the probability that conditional on =
a signal
received an agent is greater than x increasing the total proportion of agents investing at .
But then it follows that at
the mass of agents raking risky action is strictly greater than the
mass needed for a regime to change. As a consequence, it has to be the case that
decreases.
The opposite is true when x > .6
Substituting these expression into the numerator of equation (1) we obtain:
d
d
=
x
1
2
1
(
x
1=2
x
1
where the denominator is negative whenever
make the following observation:
6
1 p 1
x
x+
)+
1(
)
1
(
1=2
x =
1(
>
))
p1
2
. From the above calculations we
Substituting the expression for x into the above derivative we obtain
1=2
@
@
x
x
1
=
21+
x
x
1=2
x
1=2
x
x
1=2
2
x
This expression makes it clear that since x depends on
of the distance e¤ect.
7
(
)+
and
p
x
+
2
x
1
( )
these two parameters also determine the sign
Remark 2 The mean e¤ ect and the dispersion e¤ ects dominate distance e¤ ect. Therefore, it
is a change in x that determines how
responds to a change in x .
We are now ready to state the …rst result.7
Proposition 2 (The local e¤ect of
information. Then,
1. If
2. If
3. If
where
and
<b
=b
>b
x
x
x
( ;
x;
) then
d
d
x
( ;
x;
) then
d
d
x
( ;
x;
) then
d
d
x
b
x
( ;
x;
)=
x)
Consider a small change in the precision of private
>0
=0
<0
r
@b
x
x
x
1
+
( ;
@
x;
( ) +p
)
1
x+
1
( )
>0
Proof. See the Appendix.
The above proposition speci…es when it is the case that a marginal change in x leads to
an increase or decrease in
in terms of exogenous parameters. In particular, it states that
when the mean of the public belief is low,
< b x ( ; x ; ), then an increase in the private
precision leads to a decrease in the threshold for regime change. The opposite is true when
is high,
> b x ( ; x ; ). This result is driven by the behavior of the mean e¤ect. As
explained above an increases in x a¤ects the mean of the posterior belief of an agent by
increasing the weight the agent puts on the private signal and decreasing the weight he puts
< b x ( ; x ; ) then x is greater than
and so this shifts
on the public signal. When
agent’s posterior belief up. As a consequence, he decreases his threshold which, in light of
remark 1, leads to a decrease in . The opposite is true when
> b x ( ; x ; ).
at x
I need to check if x is greater than
4.2
Gaussian Information Structure: Global Behavior
Proposition 3 provides an easily veri…able condition under which a marginal increase in private
information leads to a decrease or increase in the regime change threshold. Note, however, that
the condition itself depends on the current precision of private and public information. This
suggests that it is possible that while an initial increase in x leads to an decrease in the
regime threshold a further change may actually increase the threshold, and vice versa. I now
investigate whether this is possible. Indeed, for some values of parameters this is the case.
Proposition 3 provides characterizations when the regime change threshold in monotone or
non-monotone function of precision x .
7
A weaker version of this result has been established by Metz (2002) and Bannier and Heinemann (2005).
8
Proposition 3 (The path of
as the function of
information that agents are initially endowed with.
<
=
>
3. Suppose that
(a) If
(b) If
(c)
x
be the precision of private
then
is increasing for all x > x ;
x
2 ( ; b ( ; x ; )) then
is initially decreasing and then increasing in
x
b ( ; x ; ) then
is decreasing for all x > x .
2. Suppose that
(a) if
(b) if
(c) if
Let
> 12 .
1. Suppose that
(a) If
(b) If
(c) If
x)
x
= 1=2.
1
2
1
2
1
2
then
then
then
is increasing for all
is constant in x ;
is decreasing in x ;
x
>
x;
< 12 .
< b x ( ; x ; ) then
is increasing for all x > x .
x
2 ( ; b ( ; x ; )) then
is initially increasing and then decreasing in
then
is decreasing for all x > x .
x
Proof. See Appendix.
Figure 1: Non-monotonicity
The above proposition states that, under some conditions,
can be a non-monotonic
function private signals’s precision. To understand why this is the case consider recall that the
direction of a change in
in response to a change in x was driven by the sign of the mean
e¤ect. Thus, to understand the behavior of
we have to understand how the mean e¤ect
changes as we keep changing x .
< b x ( ; x ; ).
For concreteness consider the case when > 12 and suppose that <
It follows that in this case, when we consider a small increase in the precision of private
information from x the mean e¤ect is negative and hence
decrease. However, a fall in
increases the mean e¤ect making it less negative. This is because a lower
implies lower x
and the mean e¤ect is driven by the di¤erence x
. As this di¤erence decreases the mean
e¤ect becomes weaker. As x increases further and further we arrive at a x keeps decreasing.
If
is not too low, i.e.
2 ( ; b x ( ; x ; )) then at one point a further increases in x
pushes x below . As a result the mean e¤ect changes sign from negative to positive. Once
this happens, a further increase in x will result in the positive mean e¤ect leading to an
increase in .8 The analogous logic applies when < 12 .
8
It is interesting to point that as mean e¤ect increases the distance e¤ect decrease. However, as shown above
the mean and the dispersion e¤ect always dominate the distance e¤ect.
9
4.3
Uniform Information Structure
Consider now the “uniform”information structure. In this case the marginal agent’s posterior
belief is simply given by
jx
unif [x
x; x + x]
We see that x a¤ects only the variance but not the mean of the posterior belief. Thus, in the
case of the “uniform information structure” the “mean e¤ect” is absent.
With the uniform information structure, the equilibrium conditions are given by:
x +
2 x
+
2 x
1
1
x
x
=
x
= 1
In order to compute the dispersion e¤ect compute x using the payo¤ indi¤erence condition:
x =
x (1
2 )
Thus, the “dispersion e¤ect” is given by
@x
=
@ x
(1
Together with the e¤ect of a change in x on
2 )
this yields:
1
(1
2 )
2
x
1+
1
2
x
Next, we can compute the “aggregate e¤ect” which is given by
1
(1
2 )
2
1+
x
1
2
x
Thus, we see that in this case the “aggregate e¤ect” and “dispersion e¤ect” cancel out and
a change in x has no e¤ect on the regime change threshold . Thus, we established the
following result.
Proposition 4 Consider a simple global game whit uniformly distributed prior belief and private signals. Then the regime change threshold
does not depend on x , the noise parameter
of private signals. It follows that
stays constant as we vary x .
The above proposition establishes that in the case of uniform information structure, a
change in precision of private signal has no e¤ect on the equilibrium threshold which stands with
a contrast to the case of Gaussian information structure. Thus, we conclude that comparative
statics with respect to parameters of information structure depend crucially on the information
structure itself.
10
5
Comparative Statics with Respect to Initial Uncertainty
5.1
Gaussian Information Structure: Local Behavior
In this section I investigate the conditions under which a marginal change in
increases and
decreases the regime change threshold . The mean e¤ect associated with a change in
is
given by
@x
x
=
@ x M ean
x+
while the “dispersion e¤ect” is given by
@x
@ x
1 1
2 x
=
Dispersion
xx
+
x+
From the above expressions we see that the “mean e¤ect”is positive (i.e., tends to increase
) when, before the change in precision of public information, the x is greater than
and
is negative (i.e., tends to decrease x ) when the x is smaller than . The intuition behind
this result is analogous to the intuition provided in section 4:1. In particular, an increase in
then the mean
increases the weight given to the public belief . When x is greater than
of the marginal agent’s posterior belief shifts down which makes him believe that, holding
constant, the regime will change with smaller probability. As a result he prefers to take action
1 rather than action 2. But this implies that the threshold signal has to increase. An analogous
intuition applies when x is smaller than .
Regarding the “dispersion e¤ect”, note that this e¤ect is exactly the same as in the case
of a change in precision of private information. In particular, the “dispersion e¤ect” depends
on the relative position of the regime change threshold and the posterior mean calculated the
critical signal. The reason behind that is simple. An increase in
decreases the variance of
the posterior belief which reduces the tails of the posterior belief. This in turn increases the
probability that >
when
is smaller then the mean of the posterior belief x xx +
and
+
decreases this probability when the opposite is true. In the …rst case, this leads to a decrease
in x while in the second case it drives x up.
Finally, note that a change in the initial uncertainty
has no e¤ect on the proportion
of agents taking a risky action holding equilibrium thresholds, x and , constant. This is
because conditional on , a change in
does not a¤ect the probability that an agent receives
a signal xi > x :
Proposition 5 Consider a small change in the precision of private information. Then,
1. If
2. If
3. If
where
<e
( ;
x;
) then
d
d
> 0;
( ;
x;
) then
d
d
=0
>e
( ;
x;
) then
d
d
<0
=e
b
( ;
x;
)=
p
1
2
x(
+
x
x+
1
)
11
!
( )
+
1
1
p
2
+
1
x
( )
and
@b
@
>0
The above proposition states that if
< b ( ; x ; ) then a marginal increase in the
precision of public information will lead to an increase in
while the opposite is true when
is low then the mean e¤ect tends
> b ( ; x ; ). The reason behind this is that when
to be positive since a low
implies a high signal threshold x . As
becomes increases the
mean e¤ect become smaller. Since the dispersion e¤ect is constant in
and therefore, there
exists a threshold
such that below this threshold an increase in
leads to an increase in
while above this threshold an increase in
leads to an increase in .
5.2
Gaussian Information Structure: Global Behavior
Proposition 3 provides an easily veri…able condition under which a marginal increase in the
precision of public belief leads to a decrease or an increase in the regime change threshold.
Note, however, that, as in the case of precision of private information, this condition itself
depends on the current precision of private and public information. This suggests that it is
possible that while an initial increase in
leads to a decrease in the regime threshold a further
change may actually increase the threshold, and vice versa. In the next proposition I describe
the global behavior of
as a function of .
Proposition 6 (The path of
as the function of ) Let
be the initial precision of
public information and
be the highest precision of public information that satis…es the
uniqueness condition.
> 12 .
1. Suppose that
(a) If
e
( ;
e
( ;
2 (e ( ;
(b) If
creasing in .
(c) If
2. Suppose that
(a) if
<
(b) if
=
(c) if
>
1
2
1
2
1
2
x;
);e
) then
( ;
x;
)) then
2[ ;
]
2[ ;
]
is initially increasing and then de-
is decreasing for all
= 1=2.
then
is increasing in
then
is constant in
then
is decreasing in
e
( ;
e
( ;
(b) If
2 (e ( ;
creasing in
(c) If
x;
is increasing for all
for all
for all
for all
2[ ;
2[ ;
2[ ;
]
]
]
< 21 .
3. Suppose that
(a) If
) then
x;
x;
x;
x;
) then
);e
) then
is increasing in
( ;
x;
)) then
is decreasing in
Proof. See the Appendix.
12
for all
2[ ;
]
2[ ;
].
is initially decreasing and then infor all
5.3
Uniform Information Structure
Recall that in the case of uniform information structure we have
unif [
; 1 + ]. Note
that in the case of the uniform information structure the level of initial uncertainty has no
e¤ect on the equilibrium play. Therefore, it follows that the regime change threshold
is
constant in .
Proposition 7 Consider a simple global game with uniformly distributed prior belief and private signals. Then the regime change threshold does not depend on , the noise parameter of
the initial prior. It follows that
stays constant as
varies.
6
A Complete Characterization of the Behavior of
6.1
Gaussian Information Structure: Local E¤ects
Above I investigated under which conditions the regime change threshold is decreasing in
precision of private and public information as well as characterized the dynamics of threshold
as a function of these precision levels. In this section I describe how these conditions are related
to each other.
In order to achieve the above goal I need to determine relation between b ( ; x ; ) and
b x ( ; x ; ). This is achieved in the next lemma.
Lemma 1 Consider b
1. If
>
1
2
2. If
=
1
2
3. If
<
1
2
Moreover,
x
( ;
then b
x
( ;
x;
x
( ;
x;
then b
x
( ;
x;
then b
x;
) and e
( ;
x;
)>e
( ;
x;
):
( ;
x;
)
)<e
( ;
x;
)
)=e
@b
@
x
>
@e
@
).
>0
Together with the previous result, the above lemma tells us that when considering the local
behavior of regime change threshold we can divide the space ( ; ) into four regions depicted
in Figure 2: (1) a region where an increase in the precision of both types of signals leads to
a decrease in
(when > 21 and
2 (e ( ; x ; ) ; b x ( ; x ; ))), (2) a region where an
increase in the precision of both types of signals leads to a decrease in
(when < 12 and
2 (b x ( ; x ; ) ; e ( ; x ; ))), (3) a region where an increase in x leads to an increase in
while an increase in leads to a decrease in ( > max f(b x ( ; x ; ) ; e ( ; x ; ))g),
and, …nally, (4) a region where an increase in x leads to a decrease in
while an increase in
leads to an increase in
( > min f(b x ( ; x ; ) ; e ( ; x ; ))g).
Figure 2 indicates also that regions where an increase in x and
have the same impact
on the regime change threshold is relatively small. For the majority states f ; g an increase
in the precision of private and public information have the opposite e¤ect. This is an important observation, as it implies that, depending on the state of the economy the government
should encourage either provision of additional private information or try to decrease initial
uncertainty but rarely do both.
13
2
1.5
1
0.5
0
-0.5
-1
0
0.25
0.5
0.75
1
Figure 2: Local variation in
6.2
Gaussian Information Structure: Global E¤ects
In this section I consider the global behavior of
and provide a similar characterization to
the one provide for local results above. Suppose that the precision of the private information
is initially …xed at the level x while the initial uncertainty is …xed at the level . Then, the
results established in Section 4 imply that for any we can divide the space of
into four
regions: (1) a region where
decreases monotonically in x and increases monotonically with
, (2) a region where is non-monotone in x and , (3) a region where in non-monotonic
in x but monotonically increasing in , and, …nally, (4) a region where
is monotonically
increasing in x and monotonically decreases in . Figure 3 depicts these regions for the case
when > 21 . For < 21 the order of the region is reversed.
As implied by Lemma 1 we can see in Figure 3 that, for any > 12 , the interval of
for
which
is non-monotonic in x is larger than the interval for which
is non-monotonic in
implying that a non-monotonicity of the regime threshold in x is a more likely phenomenon:
7
Model with Public Information
Above we considered a situation where agents have a common proper prior and have access to
private information. We argued that we can interpret the prior both as an initial uncertainty
level in the economy or the realization of public information. However, in many applications
we might allow both for a common prior, which captures history of previous play and an
explicit public signal. An additional advantage of such extension is that it allows us to perform
analysis ex-ante, unconditional on the realization of the public signal. Finally, this is the typical
14
@
@
@
@
x
< 0 for all
@
@
x
x
non-monotone in
b
@
@
non-monotone in
e
( ;
x;
x;
( ;
@
@
x
@
@
> 0 for all
x
x
)
> 0 for all
x
-
< 0 for all
)
Figure 3: Characterization of dynamics when
>
1
2
speci…cation used in the closely related quadratic-gaussian setup introduced by Morris and Shin
(2002).
Suppose that as before, agents share a common prior
N
; 1 and each of them
1
obtains a private signal xi N ; x . In addition, agents have also access to a public signal
y N ; y 1 , which is observed by all of them. Agents decide on their actions after observing
their signals.
Given that agents observe a public signal before they decided on their actions, the equilibrium regime change threshold will depend on the realization of y. In particular, de…ne a public
belief as
jy N z; z 1
where
z=
yy
+
y +
and
z
=
+
y
Then, the equilibrium regime threshold is de…ned by the critical mass condition, now given by:
r
z + x
z
1
1
( )
( )=0
(
z)
+
1=2
x
x
Thus, we see that with the public signal the mean of the common belief, z, takes the role
of the mean of the prior emphasized in Section 4. In particular, the uniqueness conditions
p
1=2
becomes now x =tz > 1= 2 . Under this assumption all the result established earlier hold,
conditional on z. However, now we can also ask questions how the change in precision of private
or public information a¤ects the equilibrium ex-ante, unconditional on z, rather than ex-post,
conditional on z (which I considered in section 4). That is, we are interested in determining
the sign of
@
@
Ez [ (z)] and
Ez [ (z)]
@ x
@ z
where Ez ( ) denotes expectations with respect to the public belief z. The main goal of the
reminder of this section is to establish conditions under which an increase in the precision of
private (or public) information leads to an increase or decrease in .
15
7.1
Technical Preliminaries
Determining how an increase in the precision of public or private information a¤ect the equilibrium in the presense of an explicit public signal is a challenging task. This is because (z)
de…ned only implicitly and the integrals such as @Ez [ (z)] =@ x that contain derivative of
normal densities are notoriously hard to compute. In this section I establish useful symmetry
and asymmetry properties of both @ =@ x and @ =@ z that will prove useful in the analysis
below. More importantly, this approach is not tied to the explicit form of @ =@ x and @ =@ z
and hence can be of interest to applied theorists working on similar models as the integrals we
are dealing here commonly arises in the setting with normally distributed random variables.
For the details we invite an interested reader in these technique to the appendix B. Here I
only state the results concerning @ =@ x and @ =@ z .
De…nition 3 A function f : R ! R is asymmetric with respect to point (a; b) if for all " > 0 :
[f (a + ")
b] 6= f (a
")
b
A function is positively asymmetric with respect to (a; b) if
[f (a + ")
b] < f (a
")
b
")
b
and negatively asymmetric with respect to (a; b) if
[f (a + ")
b] > f (a
The above de…nition generalizes symmetry of the function with respect to the origin. As
shown in the appendix, this property is extremely useful when computing the integrals where
an integrand in a product of a normal density and some function f . Next, we show that @ =@ x
and @ =@ z are asymmetric with respect to point ( x ; 0) and ( ; 0), respectively. The next
lemma states this result formally.
Lemma 2 Consider @ =@
x.
1. If
>
1
2
then @ =@
x
is positively asymmetric with respect to (
2. If
=
1
2
then @ =@
x
is symmetric with respect to (
3. If
<
1
2
then @ =@
x
is negatively asymmetric with respect to (
We have a similar result for @ =@
Lemma 3 Consider @ =@
x
x
; 0) as function of z
; 0) as function of z
x
; 0) as function of z
z.
z.
1. If
>
1
2
then @ =@
z
is positively asymmetric with respect to (
2. If
=
1
2
then @ =@
x
is symmetric with respect to (
3. If
<
1
2
then @ =@
x
is negatively asymmetric with respect to (
16
z
z
; 0) as function of z
; 0) as function of z
z
; 0) as function of z
The natural reference for the derivatives with respect to x and z is the point in which
the respective derivative intersects the zero line. The above lemmas indicate that when > 12
then the change in in regime change threshold is higher, in the absolute terms, for realization
of z above that point than those below. The opposite is true for the case of < 21 . Finally,
when = 21 then the derivatives are symmetric with respect to these points.
Finally, in order to establish the change in the expected threshold
we need the unconditional distribution of z. Since both the prior belief and the public signal are distributed
according to normal distribution it is given by:
z
N
z
;
y
7.2
(
z
+
y)
Change in the precision of private information
Having established the results regarding asymmetry of @ =@ x we can now characterize the
local behavior of a change in the expected regime change threshold, E [ ]. According to the
analysis in Section 4 we know that @ =@ x is positive when
is high while it is negative
when
is low. Therefore, we should expect that if
is high then the change in expected
should be positive when
is high and is negative when
is low. The next lemma shows
that this intuition is indeed correct.
Proposition 8 There exists a unique
1. If
>
x
then @E [ ] =@
x
>0
2. If
=
x
then @E [ ] =@
x
=0
3. If
<
x
then @E [ ] =@
x
<0
x
such that:
Proof. See Appendix.
The above result establishes that the local behavior of the regime change threshold in
carry over to the case with public information.
7.3
x
Change in the precision of public information
In this section we turn our attention to the e¤ect of an increase in the precision of public
information.
Proposition 9 There exists a unique
1. If
>
z
then @E [ ] =@
z
<0
2. If
=
z
then @E [ ] =@
z
=0
3. If
<
z
then @E [ ] =@
z
>0
z
such that:
Proof. See Appendix.
The above result establishes that the local behavior of the regime change threshold in
carry over to the case with public information.
17
x
8
Appendix
This Appendix contains all the proofs skipped in the main section of the paper.
Proof of Proposition 2. Recall from section 2 that
d
d
@
@
hx
=
x
@
+ @x
i
@
@x
1
M
@x
@ x
@x
@
P
Computing the respective partial e¤ects we obtain the following expression for the total e¤ect
of an increase in the precision of private information on :
d
d
where the sign of d =d
x
x
1
=
2
1
(
x
1 p 1
x
x+
)+
1=2
x
1
1(
)
,
1
1(
(
))
depends on the sign of the numerator:
1
(
1
)+
x
x
We …rst …nd the condition on
d =d x = 0 if and only if
1
x+
p
1
such that d =d
1
x+
p
=
x
1
( )
= 0. Keep ;
x;
constant. Then
( )
1 ( ) if and
p 1
Since
is determined by the critical mass conditin, we know
=
x+
only if the ciritical mass condition evaluated at this value of
is equal to zero. i.e.:
r
1
1
x+
1
1
1
1
p
p
(
)
+
(
)
( ) =0
1=2
+
+
x
x
x
x
or, if
r
=
x
x
1
+
( ) +p
1
x+
It follows that
d
d
= 0 i¤
r
=
x
Let
b
x
( ;
x;
)
x
1
x+
r
x
x
+
( ) +p
1
1
x+
( ) +p
1
1
x+
and note that the critical mass condition is decreasing in
and in
1 ( ) while if
x
p 1
b x ( ; x ; ) then
<
<
b
( ; x;
x+
p 1
x+
1(
) : This establishes the second part of the proposition.
18
( )
. Thus, if
) then
>
>
Proof of Proposition 3.
we obtain:
@b
x
( ; x;
@ x
)
Di¤erentiating b
1
=
2
1
( )
(
)3=2
x+
x;
"
) with respect to
r
1=2
x
x
x
1
+
x
( )
and simplifying
#
1
p
> 1= 2 ) the term in the square brackets is negative
x
( ; x;
@ x
x
@b ( ; x ;
@ x
@b x ( ; x ;
@ x
@b
( ;
1
1=2
x =
Under the uniqueness condition (
and hence
x
)
< 0 if
)
= 0 if
)
> 0 if
1
2
1
=
2
1
<
2
>
Suppose that
> 1=2. We know that as x ! 1 then b x ( ; x ; ) !
and that
x
x
b ( ; x ; ) is strictly decreasing in x . Therefore, it follows that 8 x 2 [ x ; 1) b ( ; x ; )
and that for all x > x we have b x ( ; x ; ) < b x ( ; x ; ). From these two observations
< then
< b x ( ; x ; ) for all x 2 [ x ; 1) and hence, by Proposition
we note that if
1 is decreasing in x for all x 2 [ x ; 1). Similarly, if
b x ( ; x ; ) then we know that
x
b ( ; x ; ) for any x 2 [ x ; 1) with strict inequality if x > x . From Proposition 1
it follows then that is increasing for all x > x . Finally, consider
2 ( ; b x ( ; x ; )). In
x
that case we know that
is initially lower than b ( ; x ; ) but as x increases b x ( ; x ; )
decreases monotonically towards . As a consequence, there exists a precision level bx such
that if x < bx then
is a decreasing function of x and if x > bx then
is an increasing
function of x . This establishes the …rst part of the proposition.The argument for the case
when < 1=2 is analogous.
Finally, when = 1=2 then b x ( ; x ; ) = 1=2 and hence it b x ( ; x ; ) constant in x .
Proof of Proposition 4. Recall from section 2 that
d
d
As noted in Section 5 @ =@
d
d
@
@
=
1
h
@
+ @x
i
@
@x
@x
@
M
@x
@
P
. Computing the remaining e¤ects and simplifying we obtain:
1
1=2
x
=
x
(
)+
1 1 p 1
2 x
x+
1(
)
1
(
1(
))
1=2
x
where the denominator is always positive. Therefore, we see that the sign of d =d
on the sign of the numerator:
1
x
(
)+
1 1
1
p
2 x
x+
19
1
( )
depends
We …rst …nd the condition on
if and only if
such that d =d
1
1
p
2
x+
=
But we know that
1=2
x
1
1
p
2
x+
1(
1
or
=
x;
. Then d =d
=0
( )
1
( )=0
x
1p 1
2
x+
=
1
solves the following equation:
r
x+
1
(
)+
( )
1=2
x
Thus
= 0. Fix ;
) if and only if:
r
( ) +
r
x
+
1
1
1
p
2
x+
1
( )
x
1
2
x
+
+
x
x
1
!
+
( )
x
1
1
p
2
x+
1
1
( )
=0
( )
Therefore, it follows that
d
d
= 0 i¤
r
=
1
2
x
+
+
x
1
!
( )
x
x
+
1
1
p
2
x+
1
( )
Next, note that the critical mass condition to decreasing in
and . It follows that
1p 1
1 ( ) and hence d =d
if
> e ( ; x ; ) then
<
> 0 while if
2
x+
< e ( ;
proposition.
x;
) then
( ;
@
x;
)
1
=
4(
1(
Di¤erentiating e
Proof of Proposition 6.
we obtain:
@e
1p 1
2
x+
?
1
x
+
1
)3=2
( )
"
) and hence d =d
( ;
x;
) with respect to
1
2
p
(
< 0. This proves the
+
x
+
x
)
1
x
!
( )
x
and simplifying
1=2
x
#
1=2
x
Under the uniqueness condition the term in the square brackets is negative and hence it follows
that:
@e
@e
@e
( ;
@
( ;
@
( ;
@
x;
)
x;
)
x;
)
< 0 if
= 0 if
> 0 if
1
2
1
=
2
1
<
2
>
Finally, note that uniqueness condition puts a constraint on how high the public information
p1 we let the highest
precision can be. Since
= p12 .
available to be denoted
2
20
Suppose that > 1=2. We know that as
!
then e ( ; x ; ) ! e ( ; x ; )
. It follows that for all
2 [ ; )
and that e ( ; x ; ) is strictly decreasing in
e ( ; x ; ) e ( ; x ; ) and that for all 2 ( ; ] we have e ( ; x ; ) > e ( ; x ; ).
< e ( ; x ; ) then
From these two observations we know that if
> 12 and
<
e ( ; x ; ) for all
2 [ ; ] and hence, by Proposition 5,
is increasing in the precision of public information for all
2 [ ; ]. Similarly, if
> e ( ; x ; ) then
2 [ ; ] and hence
> e ( ; x ; ) for all
is decreasing in the precision of public information for all
2 [ ; ]. The argument for the case when < 12 is analogous.
Finally, consider
2 (e ( ; x ; ) ; e ( ; x ; )). In that case we know that for
close
to ;
increases e ( ; x ; ) decreases monotonically
is lower than e ( ; x ; ) but as
towards e ( ; x ; ). As a consequence, there exists a precision level b such that if
<b
then is a decreasing function of and if
> b then is an increasing function of . This
establishes that when
2 (e ( ; x ; ) ; e ( ; x ; )) then
is a non-mnotone function
of .The argument for the case when < 1=2 is analogous.
Finally, for the case when = 12 note that in that case e ( ; x ; ) = 21 and is constant
in x .
References
Angeletos, G.-M., and A. Pavan (2007): “E¢ cient Use of Information and Social Value of
Information,” Econometrica, 75(4), 1103–1142.
Carlson, H., and E. V. Damme (1993): “Global Games and Equilibrium Selection,”Econometrica, 61(5), 989–1018.
Colombo, L., G. Femminis, and A. Pavan (2012): “Information Acquisition and Welfare,”
mimeo Northwestern University.
Corsetti, G., A. Dasgupta, S. Morris, and H. Shin (2004): “Does One Soros Make
a Di¤erence? A Theory of Currency Crises with Large and Small Traders,” Review of
Economic Studies, 71(1), 87–113.
Corsetti, G., B. Guimaraes, and N. Roubini (2006): “International lending of last resort
and moral hazard: A model of IMF’s catalytic …nance,” Journal of Monetary Economics,
53(3), 441–471.
Dasgupta, A. (2007): “Coordination and Delay in Global Games,” Journal of Economic
Theory, 134(1), 195 –225.
Eisenbach, T. (2013): “Rollover Risk: Ine¢ cient But Optimal,” working paper.
Frankel, D., S. Morris, and A. Pauzner (2003): “Equilibrium Selection in Global Games
with Strategic Complementarities,” Journal of Economic Theory, 108(1), 1–44.
Iachan, F., and P. Nenov (2014): “Information quality and crises in regime-change games,”
Journal of Economic Theory, forthcoming.
21
Morris, S., and H. Shin (1998): “Unique Equilibrium in a Model of Self-Ful…lling Currency
Attacks,” The American Economic Review, 88(3), 587–597.
(2002): “Social Value of Public Information,” American Economic Review, 92(5),
11521–1534.
(2003): “Global games: Theory and applications,” in Advances in Economics
and Econometrics: Theory and Applications, Eighth World Congress (Econometric Society Monographs) (Volume 1), ed. by M. Dewatripont, L. P. Hansen, and S. Turnovsky, pp.
56–114. Cambridge University Press.
(2004): “Coordination Risk and Price of Debt,” European Economic Review, 48,
133–153.
(2006): “Catalytic …nance: When does it work?,”Journal of International Economics,
70(1), 161–177.
Rochet, J.-C., and X. Vives (2004): “Coordination Failures and the Lender of Last Resort:
Was Bagehot Right After All?,” Journal of European Economic Association, 2(6), 1116–
1147.
Szkup, M., and I. Trevino (2009): “Information Acquisition and Transparency in Global
Games,” working paper, UBC.
Veldkamp, L. (ed.) (2011): Information Choice in Macroeconomics and Finance. Princeton
University Press.
22